COMBINATORIAL PROOF FOR THE GENERALIZED SCHUR IDENTITY

Let λ be a partition with all distinct parts. In this paper we give a bijection between the set $\Gamma$$_{λ}$(X) of pairs (equation omitted) satisfying a certain condition and the set $\pi_{λ}$(X) of circled permutation tableaux of shape λ on the set X, where P$\frac{1}{2}$ is a tail circled shifted rim hook tableaux of shape λ and (equation omitted) is a barred permutation on X. Specializing to the partition λ with one part, this bijection gives a combinatorial proof of the Schur identity: $\Sigma$2$\ell$(type($\sigma$)) = 2n! summed over all permutation $\sigma$ $\in$ $S_{n}$ with type($\sigma$) $\in$ O $P_{n}$ . .